Lemma. Let $(F,\xi),(G,\zeta):\mathcal{D}\rightrightarrows\mathcal{D}'$ be triangulated functors between pre-triangulated categories. For each object $A\in\mathcal{D}$, suppose given a morphism $\eta_A:F(A)\to G(A)$ and denote $\eta=\{\eta_A\}_{A\in\mathcal{D}}$. The following are equivalent:

1. $\eta:F\Rightarrow G$ is a natural transformation and we have a commutative triangle of natural transformations

$$\require{AMScd} \begin{CD} F\circ [1]@>{\xi}>>[1]\circ F\\ @V\eta_T VV@VVT'\eta V\\ G\circ [1]@>>\zeta> [1]\circ G \end{CD}$$

1. For every distinguished triangle $A\to B\to C\xrightarrow{h} A[1]$ in $\mathcal{D}$, the diagram $$\require{AMScd} \begin{CD} F(A)@>>> F(B) @>>> F(C)@>>> F(A)[1]\\ @VVV@VVV@VVV@VVV\\ G(A)@>>> G(B) @>>> G(C)@>>> G(A)[1] \end{CD}$$ commutes, where the vertical maps are given by $\eta_A,\eta_B,\eta_C$, and the last top and bottom arrows are $\xi_A\circ F(h)$ and $\zeta_A\circ G(h)$, respectively.

Succintly, $\eta$ is a transnatural transformation (a 2-morphism in the category of categories with translation) if and only if it is a trinatural transformation (a 2-morphism in the category of pre-triangulated categories). I wrote the proof in this question (see the Lemma at the end).